The Impulse Response Function and Ship Motion

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THE IMPULSE RESPONSE FUNCTION AND SHIP MOTIONS

W. L Cummins

AERODYNAMICS

For presentation at the

SYMPOSIUM ON SUP TREQRY

Institut fUr Schiffbau der UniverSitt Hamburg Hamburg, Germany

25 - 27 January 1962

THE IMPULSE RESPONSE PUNCION AND SHIP MOTIONS

by

W. E. Cunnnins

For presentation at the

SYMPOSIUM ON SNIP THEORY

Institut für Schiffbau der Universität Hamburg Hamburg., Germany

Just over a decade ago, Weinbium and St. Denis1 presented a compre-hensive review of the state of knowledge at the end of what we may call

the "classicalt' period in research on seakeeping. Soon after, St. Denis

and Pierson3 opened the "modern" period (some would prefer to call it

the "statistical" period) The studies of the former period were

pri-marily concerned with sinusoidal responses to sinusoidal waves, but the introduction of spectral techniques opened the door for the discusion of responses to random waves, both. long and short crested. The construction of the spectral theory on regular wave theory as a foundation deligbted

us all,, as it presented an apparent justification for the admittedly

artificial studies of the "classical"pperiod.

The activity during this last decade has been spectacular, with fiv-e major and many minor facilities for seakeeping research being opened.

Hundreds of models have been tested, many full scale trials have been run, and there has even been some real growth in our knowledge of the subject.

In particular, the spectral tool has been sharpened and tempered by the

mpricistS, and the analysts have made important advances with the rather

frightful boundary value problem. In fact, we have all been forging ahead so rapidly, that we appear to have forgotten that we are wearing a shoe which doesn't quite fit. The occasional pain from a misplaced toe is

ignored in our general enthusiasm for progress.

The "shoe" to which I refer is our inathei4tical model, the.forced

representation of the ship response by a system of secd order differential

equations. The shoe is.squeezed on, with no regard for the shape of the foot. The inadequacy of the shoe is evident in the distortions it must take if it is to be worn at all. I am referring, of course, to the fre-quency dependent coefficients which permit the mathematical model to fit the physical model, (if the excitation is purely sinusoidal, that is).

But what happens when we dont have a well defined frequency? The

mathematical model becomes almost meaningless. _{True, a Fourier analysis}

of -the exciting force (or encountered wave) _{permits the modeltô be}

retained, but physical realit _{is almost lost-in the infinit' of equations}

req.iired to represent the motion. _{The all important intuition of the engineer Ic} heavily encumbered with this grotesque baggage.

Let us consider this mathematical model briefly, and restrict ourselves

to a single degree of freedom. _{To becompletely fair, iet.us consider a}

pure, sinusoidal oscilation. _{The forcing function (if the sytem is linear)} will be sinusoidal, and can be b4ken into two components, one in phase with

the displacement and one 900 out of phase. _{We further divide the in-phase} component into a. _{restoring force, proportional to the displacement, and a} remainder. _{The latter, we call the inertial force,}

and treat it-as if it

were proportional to the instantaneous acceleration. _{The out of phase}

component, which provides all the damping, we treat as if it were proportional

to the instantaneous velocity.

We can now write an equation, which has the appearance of a differential

equation, relating these various quantities:

+ b(w)cc + c.(w)x- F sin w(t-i-)

But a differential equation is supposed to relate the instantaneous values of the function-iàvoigêd. If the periodic motl.on continues, this condition

ssatisfied.Of, course, it could just as well be satisfied by the-equation

bk + (c-auP)x = f(t)

or, more generally

(a+d) + bx +

(cZIuP)x

= f(t)

where d is arbitrary. _{These are all.equally valid models.}

One of them sto be preferred only if it truly relates _{the displacement and its first}

and second derivatives to the excitation. But suppose f(t) were to suddenly be doubled. Would the instantaneous acceleration be given by.

2f(t)-b.(W)*-C (w)x ₉

x=

a(w)

in general, no Or suppose the amplitude of the oscillation to be suddenly increased. Would the out of phase component of f(t), inediately after

the change, beequai to b*? Again, in general, no Thus, at best, b(w)

must be considered as a sort of "apparent" damping coefficient, a(w) is

an "apparent1' apparent mass, and the physical significance of both is obscure. When the oscillation consists Of several coupled modes, the

so-called coupling coefficients are equally confused and confusing.

If one restricts hImself to a phenomenologiCal investigation of how

a given ship behaves in a given wave system, these difficulties do not concern us. We simply measure responses to known waves. Most of the work

overtlie past decade has been of this nature, and much of it has been

exceilent However, sooner or later, we are required to consider not

"what,t'

but "why," and a mOre analytical technique is demanded. The phenomenological

study can tell us the effect of a change in ship loading on seakeeping

qualities only afterwe have measured it; there is no basis for quantitative

prediction given the results for one gyradius. And tie effect of achange in form i presented as an isolated result, unrelated and unrelatable to

the geometric parameters involved. We are driven to the use of the ôdél discussed

above,itl an attempt to clarify the relation of cause and effect. However; such a poor mirror of reality is of little value, and in fact can do mitch

hàt.

I am not the firstto raise this issue. . The difficulties are well

known, and a number of writers have discussed them. in particular, Tick has vigorouslY argued against our usual practice,

and has. proposed a model

which is very close to the one which will be exhibited here. His case is based solely upon the general characteristics of linear systems, while we Shall take advantage of the principles of hydrodynamiS to tie the model

to the phenomena.. More recently, Davis4 has proposed a rational approach particularly

since it was the spectral theory of statistics which at first _{gave weight}

to the investigation of responses to periodic waves.

Briefly, the specific objectives of this paper are:

Th exhibit a model. which permits the representation of the

response of a ship (in ix degrees of freedom) to an

arbitrary forcing function. (with excitation in all six modes). The model. will not involve frequency dependent.parameters.

to separate the various factors governing the response into clearly identifible units, the effect of each to be separately

determinable. Thus the effect of gyradius will be separable

from added mass. The added mass will be related only to

inertial forces and moments. The nature of the damping force

will be exhibited. The effect of coupling will be d'rivable, and the effect of "tuning" upon coupling will _{e determinable.}

In this paper we shall not consider the complementary prollem. of the relation of the exciting .force to the incident _{wave sstem. This probem}

is equally basic, and when ithas been adequetely treated, _{we will begin} to have a satisfactory framework for the interpretation of our empirical

THE IMPULSE RESPONSE FUNCTION

The basic tool whicI. will be used in this study is an elemcrtary one.,

widely used in other fields, and well known to all engineers; the imuIse

response function. It is difficult to understand jts neglect in our field.

Perhaps as Tick suggests, it is because waves look sinusoidal.

For any stable linear system, if R(t), the response to a unit impulse, is known, then the response of the system to an arbitrary otce

= (t-'r)f(T)dT

or

x(t) = R(1)f(t-T)dT

'Jo .

The only asSimptiofl required (aside from convergence) is linearity. El th

-present context this is, of course, a very strong assumption, and the purtests

will argue that it implies a thin ship or the equivalent? However all

experimental data indicate that the assumption is a good working approximation

for small to moderate oscillations of real ships fos We shall hyothesize

that.the assumption holds absolutely. V

Let Ci = 6) be displ4cements in the sx modes of response:

x1 = surge (positive forward)

= sway (positive to port) V

x = heave (positive upward)

x4 roll (positive, deck tO starboard) = pitch (positive, bow downward)

x6 = yaw (positive, bow to port)

Let R..(t) be the response in mode j to a uni-t impulse a

Note that R..() does not necessarily equal zero, though

13

which is not unstable, it will ordinarily be finite. In

f(t) is c [I]

[

L t t 0 in mode i.. in a dampe4 system: modes without a

Thus, the matrix.

_{IR .(t)I}

completely characterises the response of the ship.

.1

4-?__

-to an arbitrary excitation.

,ILI

A

i-e

Before we go on, let us onsider the relation Of these functions to

the usual coefficients. First consider the case where the modes are

uncoupled. Let

'f(t) = F _{cos(Wt + e) (3]}

where s' a phase angle whose value will be assigned later.

R(T) cos[(t-T))4T

=

F[cos(wt4)

cos uyrdr

-i- sin (wt)

$

R sin WTdT]

= [jc.

w) cos wt.) +

(w) sin (wt+cj)] (41

where

(w) ₌

_S0

COS'WT4T [..5a]

and

(u)

= Rk. (i)

sin w'rdi

restoring force (sway, surge, and yaw) the impulse response will

asymptot-ically approach some va]ue. For other modes, Ri() = 0'.

If the [f(t)}are a set of arbitrary forcing functions, the corresponding

responses are

are the Fourier cosine and sine transforms of R.11(t). We shall call

these transforms the frequency response functions., We make the further reduct ion x(t) = F. [(RC. + R. sin .). cos wt

I

11 1 Taking = tan _{1] ii} Also

x(t) =

F.[(RY.)2

+

R)2P

cos wt ) 11 f (t) cos wt - sin wt)

F.

c

cc,i

[8] i -

[(R)2 +

_(R)2]2

Now consider the usual representation

a.. + b.. + c.x. =

f(t)

1].

11

11

1

[9]

Using the x and f. from [7] and [8], it is easily seen that

+ (Re. .

RC

_{in c.)}

sin wt] 11

l

1] 1 a.

= 7 L 1

(RC ) + (RS ) ii ii R 11 R 11 =

_w[(R)

+ [lob]

Jos2

£2?YZ_

we have

tLt

-"A

J

.j

[10 a]

2

iT

Amore

useful relationship is obtained by setting = 0 in (4]:

x(t)

....

= (w). cos wt + Ri (w sin

i

Thus

_Ri

_{and Ri ae the amplitudes of the in-phase}

_and

out-of-phase

components of the response to a unit amplitude

_forcing

_{function of frequency}

w. The

impulse

_{response fUnction s related to these functions by}

R(T)

$.Rj

(w) COswT:d

1(j

(w)

sin ur.

'dw _112]

C S

using. the Fourier inversion formulas' _{Note that and R}

are uniquely

related.. If one is

_ktown,

then by [12]

_and

[5] the other is determined.

Equation [11] can also be written

x (t) Where

R(w)

tan=

c

R(w)

ThuS, the response fdl'lows the excitation

_{by the phase tan(R /RC) and}

has thi amplitude [ (Ri)2

+ '(R) _{1. .}

,

The response for a given frequency, as determined by the pair of

functions R

R C

or alternatively, the pair [(R8 )2 +

tan -

(R/R), is a

_{mapping in the frequency domain}

_{of the unit response}

function, which is defined in the time domain,. _{As equations' [4]}

_{and [11].}

permit us to pass from either domain _{to the other, the two representation,}

are completely.'equivalent'. _{Viewed in' this way, the frequency response} funct:ion is a meaningful, useful,' cOncept. It

is only when

we try to atTibute 'a deeper meaning to it, by imbedding it_{'in a false time..,domain}

model,.

that We create confusion.

+

(R)2]

_{cos [wt - e(w]}

[13]

23

Now consider the more genera1,cOuPlLe

system, with excitations in a

singlemode of the same form as given in Equation [3].

Then

x(t)

=

F[RJ

cos (wt ₊ sin (wt +

[15]

If we consider the usual repreSeTttati0n

+ bjkij +

Cjkxj)

=

where fk(t) = 0 for k i, we can develop a system

of equations in the

unknownS, aikl bik. (The cik are assumed

known from static measurements.)

All 72 of these

unknowns

are present, in principles

except where modes are

uncoupled. To determine them, it is necessarY

to consider the responses to excitations in each of the modes, separatelY.

We then have enough equations,

if we separate the in-phase and

out_of-phase components, to determine the coefficients. We have no need for

them

here, so we defer further discussion

until we

have

an analogous problem.

It is only significant to note that they can, in principle be determined

from the set of impulse response functionS, and therefore they contain no

information which is not derivable

from these functions.

Setting = 0 in [15), we

have

the system

= cos wt + siit wt

Thus, R and R5 are the amplitudes of the

in-phase

and

out-of-phase

3responses in

the j mode to unit amplitude excitation

in

the i

mode.

As before, and 11

..z\

RC coswtdw

a1(t)

_ii

\

R

sinwtdw

=1.rlo

_.1.i

= +

(R)21

cos (wt-e3) [16) 6 [18)

Where

tan

RSJ/RJ

[19]

We

_have

_{passed Over the question of convergence of the integrals in}

Equations [2] and [5]. _{Consistent with our hypothesis of linearity, we}

shall asste

Jf(t)J

is bounded. _{There will then be no difficulty unless} JJ'R1(T)d'I does not exist. _{Unfortunately, in three modes there are no}

restoring forces (or.else they are negative), and _{evidently some care is}

needed in treating these cases.,, A negative _{restoring force implies an}

unstable

system,

_{whichwould be beyond the scope of this analysis. However,} the case in

which

_{ápproaéhes some non-zerO but finite limit can be}

treated. _{The divergence of the integrals can be overcome, if we arbitrarily}

assign a value to x(0) We can formally write

x(t)

₌

R(t-T)f(T)

_{dT )}

_{R(-T)f(T) dr+}

or

(t)

_(T)fi(tT)

d

+

£ [a(tT)

-

R.(T)]f(-T)

dT

+ Xj(0)

[20]

The second integral converges, .so this expression provides _{a usable}

definition of x(t). Now let f(t) = cos wt. After an integration by

parts, We

have

x(t)

Sin w(-T)

dT

£

[R(tT)

- R1 (fl cos u

dT +

(q)

Our only concern is with the oscillatory compànents of Xj. These are

easily determined by considering the asymptotic

_form

_{of the above expression}

as t becomes.large.

_R(tT)

. and the second integral becomes

then

liin

5

x(0)

₌

o

LR(tT)

a

x(t)

_=;

-

Ri(T)J COS

wtd

()

=

j 'J

cos wt sin wt)

where Ri and are the sine and cosine transforms of R.(t). We know

that x (t) is sinusoidal, with frequency w. Therefore, this expression

holds, not only, for large t, but for all t.

If we define

R1 =_R/w

[22a]

R = [22b1

then [16] still holds. Note however, that and Ri are no longer trans-forms of Rij because these do not exist. Nevertheless, an inversion is

sill possible. Consider I

[R ('r)

- R()] cos

w1 dT

1#0 ii

= - [Ri(T)_Rjj(a)I sin (UT -

!

R (T)

sin wr dT

0

W'O

jj

= -

=

That is,

Ri is the cosine transform of [Ri(t) Rij()] and

Rij(t) =

R1() +

SO R cos wt dw

Letting

t equal zero,

RC _dw

R(a)

₌

_rr'o

_ij

[21]

so

Ru (t) =

5

R (cos wt - 1) dw

When R(o)

= 0, this reduces to [17a].

Similarly,

[a(T)

R(cD)] sin wi- di

I = -R (CD) _{/w +} 30 Ru COS WT dr

= [Rc

- R (CD)]/w ii ii Ruj(t) = R(CD) [R18 ] sin WT dw CD R(CD) [1 2

_S

sin wi dw 1

cR

sin wi dw = 0 W J W'10 ij CD

sinwtdw

1T 0 ij since LCD sin wt w dw = IT 2

Therefore, E17b] holds even when 0.

If R1 _{and Ri} are

_known,

_{it is not difficult to determine whether or}

not R(CD) = 0. _{Equation [23] gives R&) in terms of R}

[24]

CD

0

Also Tb Urn

2j

R5 sinwtdw

R(cx') 11 0

jj

c1

-urn al

_as

SflUtdw

t-'°rr'o ij w 1 im = = (O- 0 wR

using a well known theorem in Fourier transform theory (Reference 5, page 12).

2.5

When the matrix of impulse response functions is known, our first

objective of finding a representation of the ship response which is free

of frequency dependence is achieved. These functions, which we shall

collectively call the impulse response matrix, can in principle be determined experimentally.. We shall discuss such an experiment, but we defer these

remarks until we have made some progress toward our second objective.

EQUATIONS OF MOTION

/ _{The transient response of a ship has been considered by Haskind,b who} attempted an explicit solution of the. boundary value problem. This ,' we

shall not try, as we are concerned only with finding an appropriate form for the equations of motion to _{use as 'a basis for the interpretation of}

experimental results. We do not agree with certain of Haskind's hypotheses,

and. our resulting equations differ from, his in several _{important respects.}

Our approach is quite different, depending much more upon physical argument

than upon mathematical. analysis.. '' '. . . ...

Golovato7 carried out _{an experimental investigation; of the declining} oscillation, motion in pitch.. _{We shall see later that this is equivalent} to measuring the response. ,to an impulse _{However, Golôvato was not aware}

of the equivalence between the transtent and Steady _{state responses which}

we have. just. discussed,. so he attempted only to match. the coefficients derived from the transient expe;.iment _{at the frequency of the declining} oscillation, with those front a forced oscillation experiment at this frequency. _{He was handicapped because of the anomalous behavior of the} curve of declining amplitudes. For 'a simple harmonic' oscillator, this

curve isa straight line when plotted on semi-logarithmic paper. _His

curves departed. radically from Such a pattern. _{Re recognized that this} implied, that the mathematical model was faulty,. and attempted.; _{with some} success, to fit his results. with forms based on Haskind's study.

More recently, Tasai8 has performed declining oscillation' experiments in heave., using two dimensional fOrms. _{His results are not significantly} different from those of Gblovato. He matched his results at the 'measured frequency with. Ursell's theoretical results for forced oscillation. The agreement is quite good..

Just as the response of' a stable, linear, dynamic system to _{an arbitrary} force' can be. given in terms.of the dynamic response to _{an impulsive force,} the response. of our' hydrodyiiamic system _{can be stated in terms of the} hydro-dynamic response to an' impulsive displacement. _{3efore we demonstrate this}

.1 principle in the most general case, let.us.verifyit in the simplest case

j, 2CA$E I NO FORWARD SPEED

Let-the ship be. fipating at rest in st.l water. We use a. system of

coordinates .(, ., ),fixed in space, with origin i the free surface

above the center..of graytty of.the .. ,.

At time t = 0, we suppose the ship to be given an impulsive displace-ment in the 1th mode, throug the displacement c1. Th time history .of

this impulse is not significant, but. for purposes of visualzatiofl, it may

be considered tconsiSt of a movement at a large, uniform, velocity V1 for a small timet, ith the motion terminated abruptly at the end of this

time interval. Then. ,.

AX1

v1At.

During the impulse, the flow will have a velocity potential which is proportional 'to the instataneOUs impulsive

therefore, be written where is a

flow..

will

satisfy the .condtions t#P

d& /(4_..4_ cs---'.

on Ca

'i

.

4tOw

4i

/n

= 5j

on

velocity of the ship. It may,

normalized potential for impulsive

b1

jd3 )

= 0 [.26] where

Si =

' T i..= 1, 2,.3

=r

a

1j3

[.281 S [27]

S = surface of the ship

= outwardly directed unit normal

-. th

i = unit vector in j direction

= position vector with respect to c.g. of ship

It is well known8 that the above problem is equivalent to that obtained by reflecting S in = 0, and taking the surface condition over the reflection

to be the negative of that over S. _{The solution to the Neuman problem for} the flow outside this ste surface is also the solution to the given

problem in the lower half-space. _{For non-pathological Surfaces, the}

solution exists, and in fact can be computed by means of modern, high-speed

equipment.1°

During the impulse, the free surface will be elevated by _{an amount}

= - v t = - _[29]

t.f

After the impulse, this elevation will'dissipate in a radiating disturbance of the free surface, until ultimately the fluid is again at rest in the

neighborhood of the ship. _{Let the velocity potential of this decaying wave} motion. be qj(t)xj. It must satisfy the initial conditions

[30] and Xj ₌ = -g

& on

= 0 or

q1(Ci,,O,to)

= -g C3 [311

Afterward, it satisfies the usual free surface condition,

2

+g

=0

and the boundary condition on S,

an

We may take this to hold on the original position of S, introducing errors

of higher order in

only.

This is a classical problem of the

Cauchy-Poisson type, and there exists an extensive literature on the subject.

With

condition

[331,

it is more difficult, by an order of magnitude, than the

Neumann problem.

Nevertheless, it has a well defined solution.

th

- Now

let the ship undergo an arbitrary small motion in the j

mode,

x(t).

To the first order, the velocity potential of the

resulting flow

will be simply

t

\

=

[34]

It is evident that the boundary condition on S

is satisfied on the equilibrium

position of S, as the first term provides the proper

normal velocity and

= 0 on this boundary.

But also, the value of

t3/an on the actual

posLtion of S will only differ from its value on S by terms

of second and

higher order in Xj and its derivatives, so we may consider that

[34] holds

on the actual position of the hull.

To verify that the free surface condition is satisfied,

first note

that

a2®

d*

d*

acp1(0) = 4r +

c(0)

+

at

Xj

çt.

_a2cp1(t-T)

+ dT

By [261 and [30], on

= 0 this reduces to

[321

Also, or cp(0)

xi

_+J

J at

-!_tSt

cp1(t-T) i.(r) dT

-

3

Substituting these in the free surface condition

(

St

(

a21

) *.()

dT. = 0

_[35]

+

+g

by [311 and [32]. _{Thus, this condition is also satisfied, and 0 is the} required potential.

The formula [34] is a hydrodynamic analog of [1]. _{It is quite general,} and can for instance, be used to find the velocity _{potential due to a}

sinusoidal oscillation with arbitrary frequency. _{It is, of course, necessary}

to know the function

cp(t)1

_{and this presents unpleasant difficulties.}

In

this study we are content that cp(t) exists, _{and these difficulties do not}

concern us.

Of more importance than the velocity potential _{is the force acting on}

the body. _{The dynamic pressure, in our linearized -model,}

is simply p = p t p

j j

Pj(0)Ij ₊

S-ç1(t-T)

dT

acp.(t-T)

=

j$j +

(T) dT

[36] t

The net hydrodynamic force (or moment)acting on the hull in the k mode 2ç

Ct-i)

is then given by where I-,

ft

=

£

d + p S

d

\

ç(tT)

s

k

"-cp1(tT)

£00

*(T)

d1

£

Skd

d

*j(r) dT

-

;rHrjN'

= inertia of the ship in the th mode

th

= hydrostatic force in the k mode, due to

displace--' -' th

ment Xj in the .1 mode.

6jk = Kroneker delta 8jk =

1 if i

=

k,

=

0 if j

k)

t

= m1 + (t-'r) (T) dT [37] where

mjk=PS*i

skda

[38]

(-z)

= do

[39]

We can now write the equations of motion of the ship which is subjected to an arbitrary set of exciting forces, [fk(t)3. These will be

6

[(mjôjkmjkj

+ Cj lcXJ +

Kjk(tTxj(

dT]

= fk(t) [40] = _{Xj mjk}

CASE II - SHIP UNDERWAY

The case of the ship experiencing small oscillations about a reference position of mean uniform velocity is much more complex. A pair of functions,

arid p, no longer suffices, although the pattern of ouranalysis will be

similar to that followed in Case I.

We use a fixed reference system, with _{= 0 on the free surface and} with the c.g. Of the Ship at = 0 at time t 0. We suppose the ship to

be

_moving

_{with a uniformvelocityV in the direction.}

Consider the Cauchy -. Poisson problem defined by [30], [31], [32], and [331, except that now [33] is to hold on the moving surface S. This problem has a solution cp1(C1,C2,,t,v) which is, of course, 1denical withthe Ccj

of Case I when V = 0. USing this

q'j and the obtained in Case I, we may

write the velocity potential for steady motion,

where

V1(C1vt,C2,C3) +

$T,t-T

dT] [41]

= c

(Ci-VT,C2 ,C3 ,t-T)

That this satisfies the boundary condition on S is evident, as Vi1 provides the necessary instan;aneous normal velocities, and _{j/n =0 on S for' all.}

. The free surface condition :s also satisfied.,'as

may be verified by

direct evaluation,, as in Case I..

The velocity potential for tbe flow generated by the. _{ship moving with}

constant velocity, after an impulsive start at time _{zero, is}

v [ui (Cj-Vt,C

_S0

cp ('r,t-T) dl] _[421

The free surface and ship surface conditions _{are satisfied as before. The} surface elevation at t = 0 is '

or

*fj

j n =

i1 -

on S (displaced)

In three cases, solutions are immediately available. If j = 1

4. (C.-txi ,12 ,iIia) = 4r-xi

+ O(x1)

is a solution of [431 and [441, since in this case we have simple translation. Therefore [44] [44a] [45a] [

=![..V.+,(O,O)]=O

as required, and the initial conditions are met. Therefore, this must be

the stated potential.

We shall need the steady motion velocity potential for the case in which the ship is displaced by Xj from its reference position. We could,

of course, consider the displaàed ship as a completely new hull, and write down a potential similar to [41], with new functions $ and q.. Instead, we determine the corrections to the i and çj discussed above which ae necessary to satisfy the new boundary conditions. We wish a such that

*1 +

Xj = 0 on _Ca = 0 [43]

which implies that

lj = 0 [43a]

Also

-

Similarly

or

so

II,

y12-

-For j = 3, there is no such simple solution. Noting that the right side of [44a) is zero on S(original), it is only necessary to find its change

when S is displaced. Then

*i3

[45b]

*i3

-

nC3

[46]

If j = 6, the displacement is simply a rotation in yaw. The

trans-lation of a yawed body is equivalent to simultaneous transtrans-lations parallel

and perpendicular to the body axis. .Theref ore, the solution to [43] and [44] is

-(

4'

=

*1 +

-

Ci

+ *2)

sie = C

-

Ci + 1P [471

If .j = 4, or 5, the first

term

of [44a] becomes

(T3x)]

ii

The second term is

which may be written, using values of and 7* evaluated on S (original),

-

t74ri+x1(i_3Xr.7)74?i)

If we drop terms of higher order in Lx

and

use 1:27], condition [44a] reduces to .4

_-.

-4 -4

-

ij_3 X

fl

i]

[i3Xn.7*i+n(i_3X7)71Pil

or and

r ,'

si

2*i

2'Pi 1

=

-

[n.i

- -

T

- Ci [48b]

Conditions [43a) and

[44a]

are sufficient to determine Strictly,

[a1

holds on S (displaced), but we only introduce errors of order (ax)2 if we take the ship surface condition to hold on the original S. Similarly, [46] and [48) can be applied on the reference position of S.

To corresponds a with

=g

forCs=0,t=0

[49]

and with conditions corresponding to

[301, [32),

and [33] holding.

Again

we take the ship surface condition to hold on the reference position of S.

We need yet one more pair of functions. The

normal

derivative /rz

will

differ from zero on S (displaced), to the first order in AX To correct it, we define a function which satisfies the conditions

ol

= _._j.

rt

cp(T,t-T)

dT on S(displaced) [50]

4

r

(

1I1i a*1

=

_-

I

and

4t 0 on

C3=0

oj

As we do not intend to exhibit solutions for

P,

we shall not reduce

the right side. We also need a

oj' with

Poj

=

and the other appropriate conditions also holding.

We now have all the pieces needed to write the velocity potential for the flow about the ship when displaced by from its reference position.

It will be 1(T,t-T) dT] 8 = (T,tT) dT] + Ax t + 'j

{oj

q,0(T,tT) dT]

}

The 'ms

v($j +

AX1)

provide the necessary normaj. velocity in the displaced position. The normal

velocities due to

rt

v tx

oJ and V)_a, (T,t-T) dT

cancel, and none of the other terms contribute normal velocities of first order in &. Therefore, the. ship surface condition is satisfied in the

displaced position. Further, each pair of terms in brackets satisfies the

free surface condition, as may be verified by direct evaluation.

We also have all the pieces needed to assemble the potential for the

flow generated by a ship experiencing small oscillatiorm[xj(t)3. It will be

=

{ [+

(,t-)

dT]

[Xjlj+$

dT]

t

dT]}

L

j

cp(Tt-T)*(T)

[54]

The ship surface condition is satisfied as before, except now the term

provides the additional, components required for the oscillatory velocities. And again, the bracketed pairs of terms satisfy the free

surface condition.

The dynamic pres.sure at any point in the fluid is given by

6 ol

j

CiCi

I]

-

_Y

_-

x

___

p -

t

t

_Pcp1(TtT)

_(rtT)

]

dT

(It

_cp1(T,tT)

}

+ dT 1 1

cpx(T,tr)

V2 _{i_co} t

dT)

[55]

There are two convolution integrals in [55], one involving the

oscillatory displacement and one involving the oscillatory velocity. These

may be reduced to one by means of an integration by parts. We can go

so that

CT0

_1cpj1

cp41 L +

-r11

cp01 1

(0)

1

L+

J

x(i-)

dT =

x(t)

J_ci, O)

_$

(t_T)

='?

Ci

-Equation [55] now reduces to

+

_.t"

+

(

j

oJ

i)

x

V2

j

oci

+S:(cP(Tt_T)

- V

(t-'r)

₎

j

(.)

dT}

C

1çt

_vpj(T,t-T)

dT)

are concerned with the oscillatory value of the hydrodynamic

not steady components.

_{The last term [59] does not involve the}

However, when we integrate the pressure over S, the fact that

S

p

(i) dT

[60]

force, but

[XJ).

is changing

T

[cp(Tt-T) + cp(Tt-T)] dT

= cZ(tT0)

_[56]

The significance of this function cD

_{can be seen by rewriting the potential}

f or the uniform flow with the body deflected (Equation [53)).

_{It becomes}

{

+

(,t-) dT+&[

[59)

and

t

[57]

dT

[58j

its position in a steady flow field implies that even thi1s. term contributes to the oscillatory pressure. These pressures will be functions of the displacement, only.

Integrating the pressure over the surface of the ship., we can write the equation of motion:

th

c1x4 = Total hydrodynamic and hydrostatic force in the k mode,

JL%J _th

due to displacement in the .J mode.

Kjk(t_T) =

C(

acp.(r,tr) at

There are symmetries which reduce the number of coefficients. For instance

m.k = P .Sk d

C

= -p

Itj

-

do

If we consider the space enclosed by S, the free surface, and an infinite

hemisphere, we can apply Greent s theorem, and we find

mJk=-PS

an!i. do =

Further, if we consider the transverse symmetry of the ship, the matrix a.(t-T) \\

-v

[63] [.64]

il

[(mJoJk+mjkj

+ bjk*j + CjkXj +

K(t-T)

*j(T)

dT] = fk(t) [61] where m. and m

k are as defined in [40] and [381, and

.3 j

S5

($ljoj

)

5k d [621

Evidently, the matrix [bik) is of the same form, except that in general

bik # bkj. The matrix cik is even simpler, as surge and sway displacements provide no restoring forces, hydrostatic or hydrodynamic.

Therefore

The nlatrix[kJk(t)) is of the same form as [bik)

Equations [61], though similar in form to these developed by Haskind, aaskind found no hydrodynamic force proportional to the displacement, nor did he find the components of b due to and He also found that b33 = b = 0, and b = - b. The presence of in the definition of

bik makes it unlikely that such relations hold here. Further, his kernal

in the convolution integral must differ from that found here. The reason for these differences is that Haskind neglected terms in satisfying the

oundary condition on the displaced S which are of 4rst order in

Xj.

With equation [61], we have advanced a long way toward the second

objective of this paper. The dynamics of the body have been separated from

the dynamics of the fluid. Further, the hydrodynamic effects have been

[Cik) = 0

0

C31 0 c51 0 0 0 0 C43 o C82

0

0 C33 0 c53 0 0 0 0 C44 0 C64 0

0

C35

0

c55 0 0 0 0 C46 0 C68 [66] jk3 reduces to [miki = m11 o i o '115i O 0 0 1fl42 0 11152 11113 0 fl15 0 11153 0 0

fl4

0

11144 0 11154 11115 0 D15 0 fl15 0 0 0 111 0 [651

separated into separate, well defined, components, each of which can be found (in principle) from the solution of a Neumann problem or a Cauchy

-Poisson problem. Specifically, we draw the conclusions:

The equations of motion are universally valid, within the

range of validity of our assumption of linearity. That is

any excitation, periodic or non-periodic, continuous or discontinuous, is permissible, just so it results in small displacements from a condition of uniform forward velocity. The case of motion with a negative restoring force, or at

least the early history of such motion, is not excluded.

The inertial properties of the fluid are reflected in the

products m.kx.. The coefficients are independent of frequency

and of the past history of the motion, so they are legitimate

added masses. Further, they are independent of forward velocity.

There is an effect proportional to which accounts for some

of the damping. This effect 'anishes when the mean forward

speed is zero.

)

There is a hydrodynamic "restoring" force (it may be negative). It is equal to the difference between the hydrodynamic forces acting on the ship due to the steady flow in the equilibruim position and the deflected position

The effect of past history is embedded in a convolution integral over *(t). For sinusoidal motions, this integral will ordinarily have components both in phase with the motion

and 900 out of phase. The latter component contributes to the damping.

L71. _{HYDRODYNAMICS OF THE IMPULSE RESPONSE FUNCTION}

We now have two systems of relations between the excitationand the response of the ship, the impulse response relations, [2], and the

equa-tions of motion, [61]. The former are of greater value in describing the response to a given excitation, while the latter are useful in analysing

the nature of the response. Both systems hold for small oscillatory

mo-tions, so there are relations between them. We shall examine these.

First, let us start with the equations of motion, and derive the

functions fR (t)i. Suppose a ship, moving at constant forward velocity, to be subjected to a unit impulse in the I mode at time t = 0. During

the impulse, the equations of motion reduce to

k = i 6ii

where 8k is the Kroneker delta. Suppose the impulse acts during time

tt. Then, since c.

t=b*.=R (+0)

3 3 y we have 6 "k Rik

(+ 0) +m R

jk ij (+ 0) 8ik

As i and j range independently from 1 to 6, we have 36 equations relating

the two sets,

fmI

and [R..(0)1. If the equations of motion are known,

equations [671 fix the initial conditions from which the impulse response

functions can be determined. Conversely, if the impulse response func-tions are known, these equafunc-tions yield the apparent masses.

Immediately after the impulse, we have

and = 0(t) = x (0) + 0(t) 3 j x. = (0) + 0(t) 3 j K(T)*(t_T) dT = 0(t) 0

Therefore, considering only zero order terms in t, the equations of motion

yield:

R() +1[mik.cJ() + bik

R1(+0)] = 0 [68]

which relates the coefficients [b.k} to the accelerations [R(+0).

Now suppose the. ship to acted upon by a constant unit force in the jth

mode (we assume a positive restoring force to exist in this mode).

Then, after equilibruim is reached,

6

Y

= 6ik

jl

=

SORij

(r)dT or 6

R(T)

dT = 6ik [69] j=l

In modes without a positive restoring force there is difficulty, as there

is no guarantee that all of the coupling coefficients are necessarily zero.

Thus, c x

,

the sway force due to a yaw angle x6 will not ordinarily be

If we rewrite [61] in the form

6t

$iccr) R1. (t-T)dT = [70] j=l 6

-

[mj

_{6jk + mjk)} i(t) + bjkRjj(t) + Cjk1jj(t)] j=l

we have a Bet of 36 equations which can either be regarded as a set of

simultaneous integral equations for the kernals fICjk(T)} or a set of simultaneous integro-differential equations for the impulse response

functions, fR(t)3.

We have already seen (equation

_[16])

that if

f1(t) = COB (Vt

then

xj(t) = R COB Wt + R sin (Vt

Substituting these values in the equations of motion, we get

+ mjk)UPRIJ - bik

WRjJ -

cjRjj

cu(R K + R

K)]

(Vt

+

[mj

_{8jk + mJk)uPRi - bik} T.

_-

_{Cik R}

- w (R - R1

K)]

sin

Por any given frequency, this _{is an identity, so the net}

coefficients of

cos wt and sin wt must be _zero.

This gives us 72 equations relating the transforms IRi.,

R1.3

with the transforms

Kjk}. We have, or, equivalently, 6

{[c

6Jk+mjk)W_cjk_wKfk]R

C S

-

(bik + Kjk)w Rij} = _6ik S C C B

cn)(R _{Kik +}

Kik) 6

C = 6ik [(mj

8Jk + mk)aR

- bJkw R _{- Cik}

Rj]

j=l 6 r-i C C S S

-

w' _{(R Kik - R1} Kik) /- ij j=l 6

= w

jk + mjk)ciP R + b U) _R C

-cjkRij jk ij j=l Y{bJk+ K) U) Rjj

+

[mj

_{6jk + mjk)U? - Cjk_} K;klRij} = 0 [72bJ

Thus, instead of the integral and integro-differential equations relating [R) with [KjkI _{Equation [70],we have systems of linear}

equations relating their transforms.

The transforms of [RJ _{also yield useful variants of the relations}

already given. _{For instance, if we let} U) _{= 0, we have}

[71a1

[71b]

and we have =

4Cw

R1 dw

i(0) =-

_LJ

°?

RC dw TI o and 6

Cj R(0)

= 6ik .1=1

a more general form of 169].

Also, noting that

* (t) =

__.j'

W R (w) cos wt dw .1

ir0

ii

(t) =-

uP iT Jo

R(W)

COB wt dw

Therefore, [67] and (68] may be written

6

8jk +

mjk)5

w R dw]

[m

_{8jk +}

j=1 rr

- T °ik

[73] (74a] [74b] [75] [76]

0

Rj dw - bjk$w R

dW] = 0

'5

CONCLUSION

In the foregoing, we have presented two mathematical models for rep-resenting the response characteristics of ship. The equations of motion

are more general, as they apply to the initial stages of an unstable motion. Where the two systems are equally valid, we have relations which permit us

to pass (at least in principle) from either system to the other.

The impulse response function is certainly the better representation

for computing responses. It integrates all factors, mechanical,

hydro-static, and hydrodynamic, in the most efficient manner possible for corn-utation. However, for this very reason, it is a poor analytical tool for explaining why the ship responds the way it does, or how the response will be affected if any change in conditions occurs. For instance, models

are ordinarily tested with restraints in certain modes. A restraint in any mode will affect the impulse response function in ay coupled mode. Since the ship Is free in all modes, itis evidently imprto use these response functiotto predict full scale behavior, unless they are corrected

for the effect of such restraints.

The hydrodynamic equations do not suffer from this disadvantage. Known restraints are readily includable, and their effects determinable. Or a change in mass distribution can be treated independently of the

hydrodynamics.

It is not uncommon in model testing to

have "incompatiblet' parasitic inertias in the different modes. Thus, the towing gear may con-tribute a different mass in surge from that in heave. By means of the equations of motion, the effect of these inertias upon the motions can be

analyzed. Thus, the equations of motion provide a more powerfulanalytical tool for studying the relationship of the response to the parameters

governing that response.

We can conclude, then, that these two representations complement each other; the one for response calculation, the other for response analysis. In fact, if it is truly practicable to pass from one representation to the

a) Model experiments may be designed to obtain maximum accuracy,

rather than max1miim realism. Hydrodynamic effects should be emphasized in the design, since other effects are separately

determinable. Thus, one.should test at small gyradius, in order that the effect of the inertial properties of the body

itself will be minimized.

Restraints are permissible, if their character is fully known. Thus, rather than directly find the impulse response matrix, in its complete generality, more elementary experiments may be conducted to determine specific terms in the equations of

motion. One may restrict himself to one, two, or three degrees of freedom, and obtain results which are completely valid, when

interpreted by means of the equations of motion.

The recurring difficulty of handling modes in which the restoring force is zero or negative can be easily overcome. It is clear that an accurate experimental investigation of these modes would uncover practical difficulties analogous

to the theoretical ones we have discussed. However, the problem can easily be solved by imposing known restraints

(i.e. springs) which will restore positive stability. The effect of these restraints is readily includable in the equations of motion, it can be removed by calculation, and

the correct impulse response, free of restraint, can be determined.

REFERENCES

Weinbium, G., and St.Denis, Manley, "On the Motions of Ships at

Sea," Transactions, The Society of Naval Architects and Marine Engineers, Vol 58, 1950.

St. Denis, Manley, and Pierson, W. J., Jr., "On the Motions of Ships in Confused Seas," Trasactions, The Society of Naval Architects and Marine Engineers, Vol. 61, 1953.

TIck, Leo J., "Differential Equations with Frequency-Dependent

Coefficients," Journal of

Ship

_{Research, Vol. 3, No. 2, October 1959.}

Davis, Michael C., "Analysis and Control of

_Ship

_{Motion in a Random} Seaway," M.S. Thesis, Massachusetts Institute of Technology, _{June 1961.}

Sneddon, Ian. N., "Fourier Transforms," McGraw-Hill Book Company, Inc., 1951.

Haskind, M. D., "Oscillation of a Ship on a Calm Sea," Bulletin

de l'Academie des Sciences de l'UR.SS, Classe des Sciences Techniques, 1946 no. 1, pp 23-34.

Gblovato, .P., "A Study of the Transient Pitching Oscillations of a Ship," Journal of Ship Research, Vol. 2, No. 4, March 1959.

L

Tasai, Fukuzo, "On the Free Heaving of a Cylinder Floating on the Surface of a Fluid," Reports of Research Institute for Applied _Mechanics,

Vol. VIII, No. 32, 1960.

Weinblum, G. P., "On Hydrodynamic Masses," David Taylor Model

Basin Report 809, April 1952.

Smith, A. M. 0., and Pierce, Jesse, "Exact Solution of the _Neumann Problem. Calculation of Non-Circulatory Plane and Axially _{Symmetric Flows}

AEODY1AMI

.W.E. Cunirnins

IMPULSE RESPONSE FUNCTION AND SHI PMOTIONS.

S(.1PSTEKN4S( FOF$TTUTT

I(ORTFøR7"

DA1V MO11ATh

LISTEFØRT:/

HYDROMECHANICS LABORATORY RESEARCH AND DEVELOPMENT REPORT

THE IMPULSE RESPONSE FUNCTION AND

SHIP MOTIONS

by

W.E. Cummins

This paper was presented at the Symposium on

Ship Thecry at the Institut fiir Schiffbau der Universitt

Hamburg, 2 5-27 January 1962.

ABSTRACT

''

After a review of the deficiencies of the usual

equat.ions of motion for an

oscillating ship, two new representations are given. One makes use of the impulse

response function and depends only upon the system being linear. The response

is

given as a convolution integral over the past history of the exciting force with the

impulse response function appearing as the kernel. The second representation is

based upon a hydrodynarnic study, and new forms for the equations of motion are

exhibited. The equations resemble the usual equations, with the addition of

con-volution integrals over the past history of the velocity. However, the coefficients

in these new equations are independent of frequency, as are the kernel functions

in the convolution integrals. Both representations are quite general and apply to

transient motions as well as periodic. The relations between the two

representa-tions are given. The treatment considers six degrees of freedom, with linear

coupling between the various modes.

The Impulse Response Function and Ship Motions

W. E. Cuinmina

Introduction

Just over a decade ago, Weinbium and St. Denis) presented a comprehensive review of the -state of knowledge at the end of what we may call the "classical" period in reaeardt on

sea-keeping. Soon after, St. Denis and Pierson2) opened the

"modern" period (some would prefer to call it the "statistical" period). The studies of the former period were primarily con-cerned with sinusoidal responses to sinusoidal waves, but the

introduction of speceral tediniques opened the door for the

discussion of responses to random waves, both long and short

crested. The construction of the spectral theory on regular ways they as a foundation delighted us all, asit presented an apparent justification for the admittedly attificial studies of the "ilassical" period.

The activity during this last decade has been spectacular, with five major and many minor facilities for seakeeping re-seardi being opened. Hundreds of models have been tested,

many full scale trials have been run, and there has even been some real growth in our knowledge of the subject. In particu-lar, the spectral tool has been sharpened and tempered by the

empiricists, and the analysts have made important advances with the rather frightful boundary value problem. In fact, we

have aU been forging ahead so rapidly that we appear to have

forgotten that we are wearing a shoe whidi doesn't quite fit. The occasional pain from a misplaced toe is ignored in our

general enthusiasm for progress.

The "shoe" to whidi refer is our mathematical mode], the

forced representation of the ship response by a system of

second order differential equations. The shoe is squeesed on,

with no regard for the shape of the foot. The inadequacy of the shoe is evident in the distortions it must take if it is to be worn at all. I am referring, of course, to the frequency

de-pendent coefficients whith permit the mathematical model to

fit the physical model (if the excitation is purely sinusoidal,

that ía).

But what happens when we don't have a well defined

frequency? The mathematical model becomes almost meaning-less. True, a Fourier analysis of the exciting force (or

encoun-tered wave) permits the model to he retained, but physical reality is almost lost in the inflaLty of equations required to

represent the motion.

Let us consider this mathematical model briefly, and restrict ourselves to a single degree of freedom. To be completely fair, let us consider a pure, sinusoidal oscillation. The forcing

func-tion (if the systemic linear) will be sinusoidal, and can be

broken into two components, one in phase with the displace-ment and one 90° out of phase. We further divide the in-phase component into a restoring force, proportional to the

displace-ment, and a remainder. The latter we call the inertial force, and treat it as U it were proportional to the instantaneous

1) References are listed at tim end of the paper

acceleration. The out-of-phase component, whith provides all

the damping, we treat as if it were proportional to the

in-stantaneous velocity.

We can now write an equation, whidt has the appearance of a differential equation, relating these various quantities:

a(w) +b(w)± + c(to)x F08in(un+ a). But a differential equation is supposed to relate the instan-taneous values of the functions involved. If the periodic

motion continues, this condition is satisfied. Of course, it could just as well be satisfied by the equation

ha + (c - an2) x = f (t)

or more generally

(a±d)+b*+(c+dw2)x=i(t)

where d is arbitrary. These are all equally valid models. One of them isto be preferred only if it truly relates the displace-ment and its first and second derivatives to the excitation in

some more general way. But suppose f (t) were to be.auddenly doubled. Would the instantaneous acceleration be given by

2f(t)b()ic(w)x

a (to)

In general, no! Or suppose the amplitude of the oscillation to be suddenly increased. Would the out of phase component of

f(t), immediately after the diange, be equal to ha? Again, in genera], no. Thus, at best, b (to) must be considered as a sort of "apparent" damping coefficient, a (to) as an "apparent" apparent mass, and the physical significance of both is obscure. When the osi4lInHn consists of several coupled modes, the so-called coupling coefficients axe equally

con-fused and confusing.

If we restrict ourselves to a phenoinenological investigation of how a given ship behaves in a given wave system, these

dif-ficulties do not concern us. We simply measure responses to

known waves. Moat of the work over the past decade has been

of this nature, end much of it has been excellent. However, sooner or later, we are required to consider not "whet" but "why," and a more analytical technique is demanded. The phenomenological study can tell us the effect of a change in

ship loading on seakeeping qualities only after we have

mea-sured it; there is no basis for quantitative prediction given

th results for one gyradius. And the effect of a change in form

is presented as an isolated result, unrelated and unrelatable to the geometric parameters involved. We are driven to the use of the model discussed above in an attempt to clarify the relation of cause and effect. But audi a poor mirror of reality

is of little value, and in fact can do much harm.

I am not the first to raise this issue. The difficulties are well known and a number of writers have discussed them. In parti-cular, 'fleha) has vigorously argued against our usual practice and has proposed a model which is very close to the one which will be exhibited here. His case is based solely upon the

vantage of the principles of hydrodynamics to tic the model

to the phenomena. More recently, Davis) has proposed a

ratio-nal approach from the point of view of statistics. This is sug-gestive, particularly since it was the spectral theory of

stati-stics which first gave weight to the investigation of responses to periodic waves.

Briefly, the specific objectives of this paper are:

To exhibit a model which permits the representation of the response of a ship (in six degree of freedom) to an arbitrary forcing function (with excitation in all six

modes). The model will not involve frequency dependent parameters.

To separate the various factors governing the response into clearly identifiable units, the effect of eadi to be

separately determinable. Thus the effect of gyradius will

be separable from added mass. The added mass will be related only to inertial forces and moments. The nature of the damping force will be exhibited. The effect of coupling will be derivable and the effect of "tuning"

upon coupling will be determinable.

In this paper we shall not consider the complementary pro-blezn of the relation of the exciting force to the incident wave

system. This problem is equally basic, and when it has been

adequately treated, we will begin to have a satisfactory

frame-work for the interpretation of our empirical studies.

The Impulse Response Function

The basic tool which will be used in this study is an

elemen-tary one, widely used in other fields and well known to all engineers: the impulse response function. It is difficult to understand its neglect in our field. Perhaps as Tick suggests,

it is because waves look sinusoidal.

For any stable linear system, if R (t), the response to a unit

impulse, is known, then the response of the system to an arbi-trary force f (r) is t x (t) = j' R (t - v) I (c) dv -or (1) x (t) =fR (r) f (t- v) dv.

The only assumption required (aside from convergence) is linearity. In the present context this is, of course, aery strong assumption, and the purists will argue that it implies a thin

ship or the equivalent. However all experimental data indicate that the assumption is a good working approximation for small

to moderate oscillations of real ship forms. We shall

hypo-thesize that the assumption holds absolutely.

Let Xj, (I = 1,...., 6) be displacements in the aix modes

of response:

= surge (positive forward)

x2 = sway (positive to port) = heave (positive upward) x4 = roll (positive, deck to starboard) x5 = pitch (positive, bow downward)

Xe yaw (positive, bow to port)

Let Rij (t) be tho response In mode j to a unit impulse at t = (1

in mode L Note that R1 (00) does not necessarily equal zero,

though in a damped system which is not unstable, it will

ordi-narily be finite. In modes without a restoring force (sway, surge, and yaw), the impulse response will asymptotically approach some value. For other modes, R (00) = 0.

If the {fi (r)} are an arbitrary set for forcing functions, the

corresponding responses are

2

Before we go on, let us consider the relation of these

func-tions to the usual coefficients. First consider the case where

the modes arc uncoupled. Let

fi(t) = Ficos(wt + ci) (3]

where ci is a phase angle whose value will be assigned later.

xt(t) = Ft j'Rii(e) cos(w(tt) + e)d't

= F1 [cos (WI + Ct)SRn cos wvdc

0 + sin (en + ci) j'Rt sin covdv]

= Fi (R11c (Ii)) cos (wt + et) + Ru! (w) sin (on + es))

(4]

where

where

5 =

____

Rite (m) = $ R (v) cos ortdr (5a]

Ru5 ((0)= SR (e) sin wvdt [5b) are the Fourier cosine and sine transforms of (t). We shall call these transforms the frequency response functions. We make the further reduction

x(t) = F1[(R110cosE1 + R115sine1) coatst

+ (R115 cos c, - R11' cos Li) sin ofl]

Taking tan B1 = Rn5/Ri,C (6]

we have x1 (t) = F1 [(R)2 +R11c)!J'/5 cos cat. [7]

Also

f (t) F1 (R11t cos cut R116 sin cat) (8)

((R1j)2 + (R11C)n]'/S

Now consider the usual representation

a11 + b1; + c1; = f1(t) - [91

Using the x1 and f1 from (7] and [8], it.Is easily seen that

I R-. 1

a1 = i/wI c1 " I liOn)

[ (R)t + (R11)5j

IlOb] 0) [(R11c)! + (R.1')2)

A more useful relationship is obtained by setting e1 =0 in (4):

= R(o) cos cot + Rjj(w) sincot - (11]

Thus R,c and Ru5 are the amplitudes of the in-phase and out-of-phase components of the response to a unit amplitude

forc-ing function of frequency ca. The impulse response function

is related to these functions by

2('

R (v) - 1 R1(u) coscar de

0

__$Rii5(O3)5in(01CdO) [12]

using the Fourier inversion formulas. Not that R11 c and R116

are uniquely related. If one is known, then by [[2] and [51,

the other is determined.

Equation [11) can also be written

R112 + (Rj1)91' cos (cat - a1 (cc)] (13]

tan E = Re5 (w) [14)

R110(co)

xj (t) = $ R11 (r)fi (I - v) dc. [2) i1 0

Thus, the matrix {R (t)} completely characterizes the response Thus, the response follows the excitation by the phase

The response for a given frequency, as determined by the

pair of functions R.c, or alternatively, the pair [(R1)2 + (R)5J", tan4 (R,1/R1c), is a mapping in the frequency do-main of the unit response function, whidi is defined in the

time domain. As equations 141 and [11] permit us to pass from

either domain to the other, the two representations are

com-pletely equivalent. Viewed in this way, the frequency response

function is a meaningful, useful concept- It is only when we try to attribute a deeper meaning to it, by imbedding it in a

false time domain model., that we create confusion.

Now consider the more general, coupled system, with exci-tations in a single mode of the same form as given in equation

131. Then

xj (t) = F1 [R3 cos (cot + e1) + R,5 sin (cot + a1)). (15)

If we consider the usual representation

a

(a1 + bjkij+C3kXJ) =

and

I '

where k (t) 0 for k # i, we can develop a system of equa-tions in the unknowns, sJI, b - (The 0jk are assumed known

from static measurements.) All 72 of these unknowns are present, in principle, except where modes are uncoupled. To determine them, it is necessary to consider the responses to excitations in each of the modes separately. We then have

enough equations, if we separate the in.phase and out-of-phase

components, to determine the coefficients. We have no need for them here, so we defer further discussion until we face a closely related problem. It is only significant to note that they can, in principle, be determined from the set of impulse

response functions, and therefore they contain no information which is not derivable from these functions.

SettingE 0 in (15), we have the system

x1(t)

Rjj COBcot + R38 sin cot. (16)

Thus, R1 and R115 are the amplitudes of the in-phase and out.of-phase responses in the j mode to unit amplitude

ex-citation in the th mode. As before,

R11 (t) =

1..

R.11C cos Cot dts

= R5sincotdw

= [(Rc)z + (R1B)9'hl cos (cot - a)

where tanej =

We have passed over the question of convergence of the inie-grals in equations [21 and [51- Consistent with our hypothesis of linearity, we shall assume f1 (t) us bounded. There will then be no difficulty unless flR13 ( dt does not existS Unfortunate-ly, in three modes there are no restoring forces (or else they are negative), and evidently some care is needed in treating these cases. A negative restoring force implies an unstable system. which would be beyond the scope of this analysis. However, the case in which R11 approaches some non-zero but finite limit

can be treated. The divergence of the integrals can be over-come if we arbitrarily assign a value to x, (0). We formally

write t 0 xj(t) = fR11 (t_t) f1(t) dtSRu (.c) f1 (-c) dz+xj (0) °'xj(t) = $R11 (-a) f1 (tt) d 3 + $[R1 (t + t) Ru (t)] f() dt + xj (0.) (20]

The second integral converges, so this expression provides a

usable definition of xj Ct). Now let f, (t) = cos cot. After an inte-gration 1,y parts, we have

; (t) = 1/co 5 1L (t) siflW(tT) dr

+ 5 [R1 (t + a) - Ru (a)) cot we dt + ; (0). Our only concern is with the oscillatory components of x1.

These are easily determined by considering the asymptotic form of the above expression as t becomes large. Ru (t+t)+Rjj(00), and the second integral becomes constant. if we set

x1 (0) =

-

jRjj (t + a) Ru (t)] cos cot dc

then

Xj(t) =

-.! (kf

cos cot + Rjcsinwt) (21]

where ItJ and are the sine and cosine transforms of

u (t). We know that x (t) is sinusoidal, with frequency U).

Therefore, this expression holds not only for large t but for all t

If we define

Re = -

[22a)

(22b1

then [16] still holds. Note however, that Rue and R,f are no

longer transforms of R1 because these do not exist. Neverthe-less, an inversion is still possible. Consider

j'[Rjj (*R(°°))cos wtda

=Rjj(a)_Ru(00)Isinwtl_--- kij(t)sinoYedt

0) C

0

Ru0.

That is, R110 is the cosine transform of (R1 (t) Ru (00)] and

R11(t} = R11(co) + - .$Rticcoswtdw

Letting equal zero,

-(23]

so

Ru(t) =

_L$Rif(coswt_1)dw

[24]

When R,1 (00) = 0, this reduces to [17a). Similarly,

35 [Ru (-a) - Ru (00)] sin err d-a

= R11 (00)1w + $ ucos Wt dt

0)

0

= (1111C_..R,j(oO)]/w and

R(t) = Ru(00) +

_L$[Rs

Ruoo)1sin cot dco

0)

0

[ha)

(17b1